Decaying Kolmogorov turbulence in a model of superflow

Abstract

Superfluid turbulence is studied using numerical simulations of the nonlinear Schrödinger equation (NLSE), which is the correct equation of motion for superflows at low temperatures. This equation depends on two parameters: the sound velocity and the coherence length. It naturally contains nonsingular quantized vortex lines. The NLSE mass, momentum, and energy conservation relations are derived in hydrodynamic form. The total energy is decomposed into an incompressible kinetic part, and other parts that correspond to acoustic excitations. The corresponding energy spectra are defined and computed numerically in the case of the two-dimensional vortex solution. A preparation method, generating initial data reproducing the vorticity dynamics of any three-dimensional flow with Clebsch representation is given and is applied to the Taylor–Green (TG) vortex. The NLSE TG vortex is studied with resolutions up to 5123. The energetics of the flow is found to be remarkably similar to that of the viscous TG vortex. The rate of the (irreversible) transfer of kinetic energy into other energy components is comparable, both in magnitude and time scale, to the energy dissipation of the viscous flow. This transfer rate depends weakly on the coherence length. At the moment of maximum energy dissipation, the energy spectrum follows a power law compatible with Kolmogorov’s −5/3 value. Physical-space visualizations show that the vorticity dynamics of the superflow is similar to that of the viscous flow in which vortex reconnection events play a major role. It is argued that there may be some amount of universality of reconnection processes, because of topological constraints. Some preliminary support for this conjecture is given in the special case of secondary instabilities of round jets. The experimental implications of the close analogy between superfluid and viscous decaying turbulence are discussed.